- Research Article
- Open Access

# Slowly Oscillating Solutions of a Parabolic Inverse Problem: Boundary Value Problems

- Fenglin Yang
^{1}Email author and - Chuanyi Zhang
^{1}

**2010**:471491

https://doi.org/10.1155/2010/471491

© F. Yang and C. Zhang. 2010

**Received:**11 October 2010**Accepted:**20 December 2010**Published:**29 December 2010

## Abstract

The existence and uniqueness of a slowly oscillating solution to parabolic inverse problems for a type of boundary value problem are established. Stability of the solution is discussed.

## Keywords

- Differential Equation
- Continuous Function
- Integral Equation
- Partial Differential Equation
- Unique Solution

## 1. Introduction

It is well known that the space of almost periodic functions and some of its generalizations have many applications (e.g., [1–13] and references therein). However, little has been done for to inverse problems except for our work in [14–16]. Sarason in [17] studied the space of slowly oscillating functions. This is a -subalgebra of , the space of bounded, continuous, complex-valued functions on with the supremum norm . Compared with , is a quite large space (see [17–20]). What we are interested in is based on the belief that certainly has a variety of applications in many mathematical areas too. In [15], we studied slowly oscillating solutions of a parabolic inverse problem for Cauchy problems. In this paper, we devote such solutions for a type of boundary value problem.

Set . Let (resp., , where ) denote the -algebra of bounded continuous complex-valued functions on (resp., ) with the supremum norm. For (resp., ) and , the translate of by is the function (resp., , ).

- (1)
A function is called slowly oscillating if for every , , the space of the functions vanishing at infinity. Denote by the set of all such functions.

- (2)
A function is said to be slowly oscillating in and uniform on compact subsets of if for each and is uniformly continuous on for any compact subset . Denote by the set of all such functions. For convenience, such functions are also called uniformly slowly oscillating functions.

- (3)
Let be a Banach space, and let be the space of bounded continuous functions from to . If we replace in (1) by , then we get the definition of .

As in [17], we always assume that is uniformly continuous.

The following two propositions come from [15, Section 1].

Proposition 1.2.

Let be such that is uniformly continuous on . Then .

The following proposition shows that the composite is also slowly oscillating.

Proposition 1.3.

Let . If and for all , then .

In the sequel, we will use the notations: , . means that is slowly oscillating in and uniformly for ; means that is slowly oscillating in and uniformly on .

be the fundamental solution of the heat equation [21].

## 2. A Type of Boundary Value Problem

In this section, we always assume the following: , , , , , , , , and , .

be Green's function for the boundary value problems [22, 23].

where ( ) are positive and increasing for and as .

To show the main results of this section, the following lemmas are needed. The first lemma is Lemma 3.1 on page 15 in [24].

Lemma 2.1.

Lemma 2.2.

Proof.

Apply Lemma 2.1 to get the conclusion.

Lemma 2.3.

where .

One sees that depends on only and is bounded near zero.

Proof.

The existence and uniqueness of the solution comes from Theorem 5.3 on page 320 in [25].

By Lemma 2.1, one gets the desired inequality.

where is a constant. Since and are slowly oscillating, the right-hand sides of the inequality above approaches zero as . This means that . The proof is complete.

Consider the following problem.

Problem 1.

Let , and let . We have the following two additional problems for and , respectively.

Problem 2.

Problem 3.

Lemma 2.4.

Problems 1, 2, and 3 are equivalent to each other.

Proof.

the equivalence of Problems 2 and 3 can be proved similarly. The proof is complete.

where is determined by (2.37).

One can directly test that Problem 3 is equivalent to (2.37)-(2.38).

and therefore, .

Choose such that for , . Now, set . Then is a contraction from into itself, and therefore, has a unique fixed point. Thus, we have shown.

Theorem 2.5.

Let functions , , , and be as above. Then, for small , Problem 3 has a unique solution ( ) in with and .

Let be the solutions of Problem 3 in for the functions , , , and . Set , , , and . For the stability of the solution, we have the following.

Theorem 2.6.

where depends on , , , , , , , and .

Proof.

The proof is complete.

Corollary 2.7.

Under the conditions in Theorem 2.6, the solution of Problem 3 is unique.

## Declarations

### Acknowledgment

The research is supported by the NSF of China (no. 11071048).

## Authors’ Affiliations

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